Stabilizer conjecture for twisted module stacks

Let $X$ be a scheme over an ordinary commutative ring $S$, let $p:X\rightarrow\operatorname{Spec} S$ be the structure morphism, and let $\alpha\in\operatorname{Br}(S)$. Write $\mathscr{M}\mathrm{od}^X$ for the stack of quasi-coherent sheaves on $X$, and let $\operatorname{Br}(S)$ act on such stacks by Brauer twisting. The pullback map is $p^*:\operatorname{Br}(S)\rightarrow\operatorname{Br}(X)$. Stabilizer conjecture. If $\alpha$ stabilizes $\mathscr{M}\mathrm{od}^X$, then

$$\alpha\in\ker\bigl(\operatorname{Br}(S)\rightarrow\operatorname{Br}(X)\bigr).$$

The converse to the preceding corollary would identify the stabilizer of $\mathscr{M}\mathrm{od}^X$ with the Brauer classes on $S$ whose pullback to $X$ vanishes. The conjecture is verified in various cases in the paper, including for smooth projective varieties over a field with ample or anti-ample canonical bundle.